QMF Bank Design Methodology

 ISSN: 1693-6930 TELKOMNIKA Vol. 15, No. 2, June 2017 : 636 – 645 638 present section, we discuss an algorithm to design the two channel QMF bank with no matrix inversion which influences the performance of the optimization method used.

4. QMF Bank Design Methodology

The mathematical expression for the overall system function or distortion transfer function of the alias free two channel QMF bank is expressed as [21-29]. Tz=12[H 1 2 z-H 1 2 -z] 2 where for alias cancellation,synthesis filters are defined as: F 1 z=H 2 -zandF 2 z=-H 1 -z 3 To obtain the perfect reconstruction QMF bank, the overall transfer function Tz must be a pure delay.i.e. Tz=12[H 1 2 z-H 1 2 -z]=cz -n .or.xn=cxn-n 4 This equation shows that if the prototype filter H 1 z is chosen to be linear phase FIR with nil phase distortion, then to ensure linear phase FIR constraint, the impulse response h 1 [n] of low pass prototype filter H 1 z must be symmetrical .i.e. h 1 [n]=h 1 [N-1-n],0nN-1 where N is the filter length [17]. The corresponding frequency response is given by [5] as: H 1 e jω =Aωe - jωN-12 5 where Aω= H 1 e jω is the amplitude function.If the prototype flter H 1 z characteristics are assumed ideal in pass band and stop band regions then the exact reconstruction condition is satisfied for 0ωω p and ω s ωπ. Here ω p and ω s are the passband and stopband edge frequencies. The constraint comes in the transition bandω p ω ω s where Amplitude Distortion needs to be controlled. It means our motive is to optimize the coefficients of H 1 z such that the exact reconstruction condition is approximately nearly satisfied. Design Implementation : A Matlab code has been run to implement the design part of prototype LPF and inference has been drawn on the basis of CPU processing time taken to run that event.The entire process here involves three steps mainly 1. Design Specification for Efficient Two Channel QMF Bank2. Implementation of Two Channel QMF 3. Evaluation of Designed QMF results. Case Study Design Example: We have designed the QMF here for N=24,32,42,48,64,128 with Stopband Frequency Fsb=0.359 for Alpha constant lying between 0 and 10.22,ω s =0.6π,ω p =0.4π. A Matlab code has been written which implements the design procedure for prototype low pass filter described and tested on a laptop equipped with an Intel Core i5-2410M Processor 2.30GHz with Turbo Boost upto 2.90 GHz with 4GB RAM on Windows 7 64-bit Operating system. This section presents a design example to check the effectiveness of the proposed algorithm.The main parameters which govern the performance of the algorithm are First lobe Stop band attenuation,Stop band edge attenuation, A s =- 20log 10 H 1 ω s , Maximum Overall RipplePassband-Ripple, Prototype QMF Length N, Stopband-FrequencyFsb,Roll off factor Alpha, Measure of Reconstruction ErrordB=max10logTe jω -min10logTe jω . The Design Specification: N-Number of coefficients for the lowpass prototype. f sb - Normalized stop band edge frequency for the low pass prototype, where 0.25 f sb 0.5. Alpha- Relative weighting between the stop band energy and the ripple in the overall response. An increase in alpha will lead to greater stop band attenuation in the low pass prototype. A sample filter has been designed with QMF Design 32, 0.3, 1. The performance of the designed algorithm can be analysed through a set of observations in terms of Magnitude w.r.t. Normalized Frequency π radianssample, Phase w.r.t. Normalized Frequency plots which also reveal the Phase Distortion occuring at values of Normalized Frequency. The results of the used algorithm have been compared to the results obtainable in case of Jain-Crochiere [5] and S.K.Aggarwal- O.P.Sahu [31]. The following set of filter coefficients have been obtained for 0 n N2-1 in TELKOMNIKA ISSN: 1693-6930  Design and Implementation of Efficient Analysis and Synthesis QMF Bank… A.S. Kang 639 case of Filter Jain Crochiere and subsequent set of filter coefficients obtained for O.P.Sahu et al and that obtained in case of our proposed algorithm given in Table 1,2,3, and 4. Table 1. QMF coefficients generated [Jain Crochiere] Optimized Filter Tap Weights in QMF design By Jain and Croichere [N=32]at fsb=0.3;α=1 Table 2. QMF coefficients generated {Sahu et al] Optimized Filter Tap Weights in QMF design By O.P.Sahu et al [N=24;fsb=0.3;α=1] h 1 0=0.0034 h 1 1= -0.0074 h 1 2=-0.0022 h 1 3= 0.0163 h 1 4= -0.0020 h 1 5= -0.0301 h 1 6= 0.0124 h 1 7= 0.0525 h 1 8= -0.0375 h 1 9= -0.1000 h 1 10= 0.1272 h 1 11= 0.4672 Table 3. QMF coefficients generated [Kang Vig-Proposed Algorithm] Optimized Filter Tap Weights in QMF design by Kang-Vig [N=24;fsb=0.3;α=0.22] h 1 0=0.0036 h 1 1=-0.0072 h 1 2=-0.0023 h 1 3= 0.0161 h 1 4= -0.0018 h 1 5= -0.0300 h 1 6= 0.0122 h 1 7= 0.0524 h 1 8= -0.0373 h 1 9= -0.1000 h 1 10= 0.1270 h 1 11= 0.4673 Table 4. QMF coefficients generated [Vig Kang-Proposed Algorithm] Optimized Filter Tap Weights in QMF design by Vig-Kang [N=32;fsb=0.3;α=0.22] Figure 1. N=32;Fsb=0.3;Alpha=1 [Jain Croicere] Figure 2. N=64;Fsb=0.3;Alpha=1 h 1 0=0.46513280 h 1 1= 0.13063700 h 1 2=-0.99656700E-1 h 1 3= -0.41773659E-1 h 1 4= 0.53938050E-1 h 1 5= 0.16805820E-1 h 1 6= -0.33077250E-1 h 1 7= -0.58240110E-2 h 1 8= 0.20216010E-1 h 1 9= 0.71798260E-3 h 1 10= -0.11586330E-1 h 1 11= 0.12928400E-2 h 1 12= 0.58649780E-2 h 1 13= -0.16349580E-2 h 1 14= -0.23388170E-2 h 1 15= 0.12488120E-2 h 1 0=0.0013 h 1 1= -0.0023 h 1 2=-0.0016 h 1 3= 0.0058 h 1 4= 0.0013 h 1 5= -0.0115 h 1 6= 0.0006 h 1 7= 0.0201 h 1 8= -0.0057 h 1 9= -0.0330 h 1 10= 0.0167 h 1 11= 0.0539 h 1 12= -0.0417 h 1 13= -0.0997 h 1 14= 0.1305 h 1 15= 0.4652  ISSN: 1693-6930 TELKOMNIKA Vol. 15, No. 2, June 2017 : 636 – 645 640 Figure 3. N=128;Fsb=0.3;Alpha=1 Figure 4. N=42;Fsb=0.3;Alpha=1 Figure 5. N=24;Fsb=0.3;Alpha=1 [Sahu] Figure 6. N=24;Fsb=0.3;Alpha=0.22 Figure 7. N=48;Fsb=0.3;Alpha=0.22 Figure 8. N=32;Fsb=0.3;Alpha=0.22 Figure 9. N=64;Fsb=0.3;Alpha=0.22 Figure.10 N=128;Fsb=0.3;Alpha=0.22 TELKOMNIKA ISSN: 1693-6930  Design and Implementation of Efficient Analysis and Synthesis QMF Bank… A.S. Kang 641 Figure 11. N=24,Fsb=0.3,Alpha=1 [O.P.Sahu et al] Figure 12. Magnitude wrt Frequency Plot for a Gaussian Noise Input b Gaussian Noise OutputcGaussian Noise Intermediate Signal U0d Gaussian Noise Intermediate Signal U1 Filter Length N=78 Figure 13. Magnitude wrt Frequency Plot for, a Gaussian Noise Input, b Gaussian Noise Output, c Gaussian Noise Intermediate Signal U0, d Gaussian Noise Intermediate Signal U1 Filter Length N=24  ISSN: 1693-6930 TELKOMNIKA Vol. 15, No. 2, June 2017 : 636 – 645 642

5. Results and Discussion